Continuous S-transform (ST) can be regarded as a hybrid of Gabor and continuous wavelet transforms, providing a “time frequency representation” (TFR) of a signal by localizing with a Gaussian window that depends on the frequency. Its discrete 1-dimensional form (1D ST) is finding many applications in processing signals and time series, while its discrete 2-dimensional form (2D ST) is used for processing 2-dimensional data and images, where it should be more correctly called a “space frequency representation” (SFR), as it represents the localized frequency spectrum at each point in the 2-dimensional data set or at each pixel in the image.
Fast Time Frequency Transform tools have been developed, such as a FTFT-1D and FTFT-2D (Fast Time Frequency Transform), that generate discrete 1D ST values and 2D ST magnitudes fast and accurately. The FTFT-2D can produce local ST magnitudes at each pixel in a medical image, as well as ST statistics over a region of interest (ROI) in the image. However, the discretization of 2D ST renderings are not rotationally invariant. By rotational invariance of an SFR, it is meant that when the image is rotated by any angle, the radial component of the SFR is unchanged. This is desirable as the pathology inferred from this radial component should not be affected when the patient is positioned at a different orientation on the imaging couch.